Making particles grow while sticking to an object
Use Hair type instead of Emitter in Particle settings (suits all kinds of mesh deformations including armatures), it also provides object instancing.

Or stay with Newtone option, but use Lattice + Hook combo (for simple scale/rotate/locate animation).

Prepare Lattice

Add Lattice object, scale it in object mode a bit bigger than your mesh Cube. Change interpolation to Linear.

Hooking

Go into Lattice edit mode, select all vertices and press Ctrl H, select Hook to new object (Empty will be added)

Setting emitter object

Add Lattice modifier after Particle system and select your lattice object.

Animating

Now if you try to move Empty, it will engage movement for Lattice and also emitter as it is cotrolled by lattice object. So instead of animating emitter object, animate Empty.